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We are familiar with the fact that *real* values of correspond to complex values of .Now let us look at *complex* values of :
| |
(44) |

Thus, when , |*Z*|<1.
In words, we transform the upper half of the -plane
to the interior of the unit circle in the *Z*-plane.
Likewise, the stable region for poles is the lower half of the
-plane,
which is the exterior of the unit circle.
Figure 12 shows the transformation.
Some engineering books choose a different sign convention
(),
but I selected the sign convention of physics.
**Z
**

Figure 12
Left is the complex -plane
with axes . Right is the *Z*-plane
with axes . The words ``Convergent'' and ``Divergent''
are transformed by .

** Next:** The meaning of divergence
** Up:** INSTABILITY
** Previous:** The unit circle
Stanford Exploration Project

10/21/1998